Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?
More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of 


*

*a function 
$$u_1 \in BV(\mathbb R^2; \mathbb R)$$
with only jump part in the derivative $$Du_1 = D^{jump} u_1$$

*and of a function with only Cantor part in the derivative:
$$u_2 \in BV(\mathbb R^2; \mathbb R)$$
with $$Du_2 = D^{cantor} u_2$$

A related more general question is Heuristic and graphic representation of BV functions and their singularities

Clearly one could take a one dimensional example $f \in BV(\mathbb R)$ and then consider $g(x_1,x_2) := f(x_1)$. However, I'd like to see a "genuinely" two-dimensional example (if it exists).
 A: The absolutely continuous part and the jump part of the derivative are easy to understand: taking $\mathbf 1_{\mathbf R_+}(x_1)$ provides a $BV$ function with jump at $x_1=0$. As far as the Cantor part is concerned, you just have to think about the Cantor measure, namely the (distribution) derivative of the "devil" staircase: you know certainly the Cantor set $K$ which appears as
$$
K=\cap_{n\ge 0}K_n,
$$
with $K_0=[0,1]$, $K_1=[0,1/3]\cup[2/3,1]$, $K_n$ is the union of $2^n$ compact intervals $I_{n,l}$with length $3^{-n}$ and $$K_{n+1}=\cup_{1\le l\le 2^n}\bigl(I_{n,l}\backslash {\text{its open middle third}}\bigr).$$
Then it is easy to see that the probability measures $\mathbf 1_{K_n}/\vert K_n\vert$ converge weakly toward the so-called Cantor measure which is singular with respect to the Lebesgue measure and is diffuse (no atoms). An antiderivative of the Cantor measure belongs indeed to $BV$ with a singular derivative without jumps.
A: You could take $u$ to be the indicator function of a square. Then the jump part will be supported on the boundary of the square. 
